The behavior of bounds and oscillations of sequences of functions under regular transformations

Abstract

Introduction. Much has been written during the past thirty years on summabilityt of sequences of constants. The subject of summability of sequences of functions is largely unexplored.: Very recently Gillespie and Hurwitz? discovered that any bounded sequence of real continuous functions, defined over a closed compact set A, which converges to a continuous limit function is uniformly summable by a regular transformation totally equivalent to a regular transformation with a triangular matrix. This fact creates an interest in summability of sequences of functions. In 1925 Hurwitz? discussed the behavior of what we shall call ultimate bounds of sequences of real constants under real regular transformations, and more recently hell has considered the behavior of oscillations of complex and of real sequences of constants under complex and real regular transformations with triangular matrices. In this paper, regular transformations with triangular matrices, which have been conspicuous in the theory of summability, will be applied to sequences of functions. We will consider the behavior of ultimate bounds of real sequences under real transformations, and of continuous oscillation and convergence and uniform oscillation and convergence of complex and real sequences under complex and real transformations; ultimate bounds, oscillations, and convergence being considered (1) over a set as a whole, (2) at a single point of a set, and (3) at all points and limit points of a set.**